Question

A random sample of 200 employees at a specific company is asked to state their gender and annual salary. The results are recorded in the table.

Gender	$50,000	$75,000	$100,000	>$100,000
Male	29	34	12	20
Female	41	44	6	14

A chi-square test is used to test the null hypothesis that gender and salary are independent. Which statement is correct?

Fail to reject H₀ at the 0.10 significance level. Fail to reject H₀ at the 0.5 significance level. Reject H₀ at the 0.01 significance level. Reject H₀ at the 0.05 significance level. Reject H₀ at the 0.001 significance level.

Answer 1

Solution:

A chi-square test is used to test the null hypothesis that gender and salary are independent.

Step 1) State H0 and H1:

H0: Gender and salary are independent

Vs

H1: Gender and salary are not independent.

Step 2) Test statistic:

Where

O_ij = Observed frequencies for i^th row and j^th column.

E_ij = Expected frequencies for i^th row and j^th column.

Where

thus

Gender	$50,000	$75,000	$100,000	>$100,000
Male	29	34	12	20	R1 =95
Female	41	44	6	14	R2=105
Total	C1 = 70	C2 =78	C3 =18	C4 =34	N = 200

Thus

Thus

Oij	Eij	Oij2/Eij
29	33.25	25.293
34	37.05	31.201
12	8.55	16.842
20	16.15	24.768
41	36.75	45.741
44	40.95	47.277
6	9.45	3.810
14	17.85	10.980
N =200

Thus

Step 3) P-value:

df = ( R - 1) X (C - 1)

where

R = Number of Rows = 2

C = Number of Columns = 4

thus

df = ( R - 1) X (C - 1)

df = ( 2 - 1) X (4 - 1)

df = 1 X 3

df = 3

Significance level = 0.10 , 0.5 , 0.05 , 0.01, 0.001

Use following Excel command:

=CHISQ.DIST.RT(x , df)

=CHISQ.DIST.RT(5.913,3)

=0.1159

Thus P-value = 0.1159

Decision Rule:
Reject null hypothesis H0, if P-value < level of significance, otherwise we fail to reject H0

Since P-value = 0.1159 > 0.10 level of significance , we fail to reject H0 at 0.10 level of significance.

Thus correct answer is:

Fail to reject H₀ at the 0.10 significance level.